To obtain the effective potential from the diagrammatic approach as shown in equation (REF ), we begin by generating all the relevant generic topologies contributing at different orders of \(G_N\) . It could be easily seen from the virial theorem that N\(^n\) LO has contributions from terms proportional to \(G_N^l\)  where \(l = 1, 2, ..., n + 1\) , and we consider all the topologies at \(l-1\)  loops contributing to the specific order \(l\) .
So, for the computation of the N\(^3\) LO spin-orbit potential, we generate all the topologies till the order \(G_N^4\)  (3-loop)
using QGRAF [1]}.
There is 1 topology at order \(G_N\)  (tree-level), 2 topologies at order \(G_N^2\)  (one-loop), 9 topologies at \(G_N^3\)  (two-loop), and 32 topologies at order \(G_N^4\)  (three-loop).
Then we dress these topologies with the KK field and Feynman rules derived from the action of PNThe Feynman rules obtained from the actions in equation (REF ) and (REF ) after the KK parameterization are provided in an ancillary file Feynman_Rules.m with this article. expansion of GR
given in (REF ) and (REF ), to obtain all the Feynman diagrams
that contribute to the given order of \(G_N\)  and \(v\)  depending on the specific perturbation order.
The number of diagrams that contribute at particular order in \(1/c\)  and of particular loop topology are given in table REF  and  REFWhile considering the spin effects, we count only the representative Feynman diagrams, where the spin can contribute from any of the world-line graviton interaction vertex present in the diagram.
Additionally, the diagrams, which can be obtained from the change in the label \(1\leftrightarrow 2\) , are not counted as separate diagrams..
<FIGURE><TABLE>